Cluster characters and the combinatorics of Toda systems

We survey some connections between Toda systems and cluster algebras. One of these connections is based on representation theory{:} it is known that Laurent expansions of cluster variables are generating functions of Euler characteristics of quiver Grassmannians, and the same turns out to be true of the Hamiltonians of the open relativistic Toda chain. Another connection is geometric{\rm:} the closed nonrelativistic Toda chain can be regarded as a meromorphic Hitchin system and studied from the standpoint of spectral networks. From this standpoint, the combinatorial formulas for the Hamiltonians of the open relativistic system are sums of trajectories of differential equations defined by the closed nonrelativistic spectral curves.